diff --git a/src/kernel/metavar.cpp b/src/kernel/metavar.cpp
index d3b1ad904..99bca100d 100644
--- a/src/kernel/metavar.cpp
+++ b/src/kernel/metavar.cpp
@@ -14,6 +14,11 @@ Author: Leonardo de Moura
 #include "kernel/level.h"
 
 namespace lean {
+substitution::substitution(expr_map const & em, level_map const & lm):
+    m_expr_subst(em), m_level_subst(lm) {}
+
+substitution::substitution() {}
+
 bool substitution::is_expr_assigned(name const & m) const {
     return m_expr_subst.contains(m);
 }
@@ -54,23 +59,40 @@ optional<level> substitution::get_level(name const & m) const {
         return none_level();
 }
 
-void substitution::assign(name const & m, expr const & t, justification const & j) {
+void substitution::d_assign(name const & m, expr const & t, justification const & j) {
     lean_assert(closed(t));
     m_expr_subst.insert(m, mk_pair(t, j));
 }
 
-void substitution::assign(name const & m, expr const & t) {
+void substitution::d_assign(name const & m, expr const & t) {
     assign(m, t, justification());
 }
 
-void substitution::assign(name const & m, level const & l, justification const & j) {
+void substitution::d_assign(name const & m, level const & l, justification const & j) {
     m_level_subst.insert(m, mk_pair(l, j));
 }
 
-void substitution::assign(name const & m, level const & l) {
+void substitution::d_assign(name const & m, level const & l) {
     assign(m, l, justification());
 }
 
+substitution substitution::assign(name const & m, expr const & t, justification const & j) const {
+    lean_assert(closed(t));
+    return substitution(insert(m_expr_subst, m, mk_pair(t, j)), m_level_subst);
+}
+
+substitution substitution::assign(name const & m, expr const & t) const {
+    return assign(m, t, justification());
+}
+
+substitution substitution::assign(name const & m, level const & l, justification const & j) const {
+    return substitution(m_expr_subst, insert(m_level_subst, m, mk_pair(l, j)));
+}
+
+substitution substitution::assign(name const & m, level const & l) const {
+    return assign(m, l, justification());
+}
+
 void substitution::for_each(std::function<void(name const & n, expr const & e, justification const & j)> const & fn) const {
     m_expr_subst.for_each([=](name const & n, std::pair<expr, justification> const & a) {
             fn(n, a.first, a.second);
@@ -83,7 +105,7 @@ void substitution::for_each(std::function<void(name const & n, level const & e,
         });
 }
 
-std::pair<level, justification> instantiate_metavars(level const & l, substitution & s, bool use_jst, bool updt) {
+std::pair<level, justification> substitution::d_instantiate_metavars(level const & l, bool use_jst, bool updt) {
     if (!has_param(l))
         return mk_pair(l, justification());
     justification j;
@@ -92,16 +114,16 @@ std::pair<level, justification> instantiate_metavars(level const & l, substituti
             if (!has_meta(l)) {
                 return some_level(l);
             } else if (is_meta(l)) {
-                auto p1 = s.get_assignment(l);
+                auto p1 = get_assignment(l);
                 if (p1) {
-                    auto p2 = instantiate_metavars(p1->first, s, use_jst, updt);
+                    auto p2 = d_instantiate_metavars(p1->first, use_jst, updt);
                     if (use_jst) {
                         justification new_jst = mk_composite1(p1->second, p2.second);
                         if (updt)
-                            s.assign(l, p2.first, new_jst);
+                            d_assign(meta_id(l), p2.first, new_jst);
                         save_jst(new_jst);
                     } else if (updt) {
-                        s.assign(l, p2.first);
+                        d_assign(meta_id(l), p2.first);
                     }
                     return some_level(p2.first);
                 }
@@ -121,7 +143,7 @@ protected:
     void save_jst(justification const & j) { m_jst = mk_composite1(m_jst, j); }
 
     level visit_level(level const & l) {
-        auto p1 = instantiate_metavars(l, m_subst, m_use_jst, m_update);
+        auto p1 = m_subst.d_instantiate_metavars(l, m_use_jst, m_update);
         if (m_use_jst)
             save_jst(p1.second);
         return p1.first;
@@ -151,7 +173,7 @@ protected:
                 if (m_update) {
                     auto p2 = m_subst.d_instantiate_metavars(p1->first);
                     justification new_jst = mk_composite1(p1->second, p2.second);
-                    m_subst.assign(m_name, p2.first, new_jst);
+                    m_subst.d_assign(m_name, p2.first, new_jst);
                     save_jst(new_jst);
                     return p2.first;
                 } else {
@@ -162,7 +184,7 @@ protected:
             } else {
                 if (m_update) {
                     expr r = m_subst.d_instantiate_metavars_wo_jst(p1->first);
-                    m_subst.assign(m_name, r);
+                    m_subst.d_assign(m_name, r);
                     return r;
                 } else {
                     return m_subst.instantiate_metavars_wo_jst(p1->first);
@@ -205,7 +227,8 @@ public:
 };
 
 std::pair<expr, justification> substitution::instantiate_metavars(expr const & e) const {
-    instantiate_metavars_fn fn(const_cast<substitution&>(*this), true, false);
+    substitution s(*this);
+    instantiate_metavars_fn fn(s, true, false);
     expr r = fn(e);
     return mk_pair(r, fn.get_justification());
 }
@@ -216,20 +239,35 @@ std::pair<expr, justification> substitution::d_instantiate_metavars(expr const &
     return mk_pair(r, fn.get_justification());
 }
 
+std::tuple<expr, justification, substitution> substitution::updt_instantiate_metavars(expr const & e) const {
+    substitution s(*this);
+    instantiate_metavars_fn fn(s, true, true);
+    expr r = fn(e);
+    return std::make_tuple(r, fn.get_justification(), s);
+}
+
 std::pair<level, justification> substitution::instantiate_metavars(level const & l) const {
-    return lean::instantiate_metavars(l, const_cast<substitution&>(*this), true, false);
+    substitution s(*this);
+    return s.d_instantiate_metavars(l, true, false);
 }
 
 expr substitution::instantiate_metavars_wo_jst(expr const & e) const {
-    return instantiate_metavars_fn(const_cast<substitution&>(*this), false, false)(e);
+    substitution s(*this);
+    return instantiate_metavars_fn(s, false, false)(e);
 }
 
 expr substitution::d_instantiate_metavars_wo_jst(expr const & e) {
     return instantiate_metavars_fn(*this, false, true)(e);
 }
 
+std::pair<expr, substitution> substitution::updt_instantiate_metavars_wo_jst(expr const & e) const {
+    substitution s(*this);
+    return mk_pair(instantiate_metavars_fn(s, false, true)(e), s);
+}
+
 level substitution::instantiate_metavars_wo_jst(level const & l) const {
-    return lean::instantiate_metavars(l, const_cast<substitution&>(*this), false, false).first;
+    substitution s(*this);
+    return s.d_instantiate_metavars(l, false, false).first;
 }
 
 bool substitution::occurs_expr(name const & m, expr const & e) const {
diff --git a/src/kernel/metavar.h b/src/kernel/metavar.h
index f00487fc9..dfabb356e 100644
--- a/src/kernel/metavar.h
+++ b/src/kernel/metavar.h
@@ -13,9 +13,22 @@ Author: Leonardo de Moura
 
 namespace lean {
 class substitution {
-    rb_map<name, std::pair<expr, justification>, name_quick_cmp>  m_expr_subst;
-    rb_map<name, std::pair<level, justification>, name_quick_cmp> m_level_subst;
+    typedef rb_map<name, std::pair<expr, justification>, name_quick_cmp> expr_map;
+    typedef rb_map<name, std::pair<level, justification>, name_quick_cmp> level_map;
+    expr_map  m_expr_subst;
+    level_map m_level_subst;
+
+    substitution(expr_map const & em, level_map const & lm);
+    void d_assign(name const & m, expr const & t, justification const & j);
+    void d_assign(name const & m, expr const & t);
+    void d_assign(name const & m, level const & t, justification const & j);
+    void d_assign(name const & m, level const & t);
+    std::pair<expr, justification> d_instantiate_metavars(expr const & e);
+    expr d_instantiate_metavars_wo_jst(expr const & e);
+    std::pair<level, justification> d_instantiate_metavars(level const & l, bool use_jst, bool updt);
+    friend class instantiate_metavars_fn;
 public:
+    substitution();
     typedef optional<std::pair<expr,  justification>> opt_expr_jst;
     typedef optional<std::pair<level, justification>> opt_level_jst;
 
@@ -28,11 +41,11 @@ public:
     optional<expr> get_expr(name const & m) const;
     optional<level> get_level(name const & m) const;
 
-    void assign(name const & m, expr const & t, justification const & j);
-    void assign(name const & m, expr const & t);
+    substitution assign(name const & m, expr const & t, justification const & j) const;
+    substitution assign(name const & m, expr const & t) const;
 
-    void assign(name const & m, level const & t, justification const & j);
-    void assign(name const & m, level const & t);
+    substitution assign(name const & m, level const & t, justification const & j) const;
+    substitution assign(name const & m, level const & t) const;
 
     void for_each(std::function<void(name const & n, expr const & e, justification const & j)> const & fn) const;
     void for_each(std::function<void(name const & n, level const & e, justification const & j)> const & fn) const;
@@ -40,14 +53,14 @@ public:
     bool is_assigned(expr const & m) const { lean_assert(is_metavar(m)); return is_expr_assigned(mlocal_name(m)); }
     opt_expr_jst get_assignment(expr const & m) const { lean_assert(is_metavar(m)); return get_expr_assignment(mlocal_name(m)); }
     optional<expr> get_expr(expr const & m) const { lean_assert(is_metavar(m)); return get_expr(mlocal_name(m)); }
-    void assign(expr const & m, expr const & t, justification const & j) { lean_assert(is_metavar(m)); assign(mlocal_name(m), t, j); }
-    void assign(expr const & m, expr const & t) { lean_assert(is_metavar(m)); return assign(mlocal_name(m), t); }
+    substitution assign(expr const & m, expr const & t, justification const & j) { lean_assert(is_metavar(m)); return assign(mlocal_name(m), t, j); }
+    substitution assign(expr const & m, expr const & t) const { lean_assert(is_metavar(m)); return assign(mlocal_name(m), t); }
 
     bool is_assigned(level const & m) const { lean_assert(is_meta(m)); return is_level_assigned(meta_id(m)); }
     opt_level_jst get_assignment(level const & m) const { lean_assert(is_meta(m)); return get_level_assignment(meta_id(m)); }
     optional<level> get_level(level const & m) const { lean_assert(is_meta(m)); return get_level(meta_id(m)); }
-    void assign(level const & m, level const & l, justification const & j) { lean_assert(is_meta(m)); assign(meta_id(m), l, j); }
-    void assign(level const & m, level const & l) { lean_assert(is_meta(m)); return assign(meta_id(m), l); }
+    substitution assign(level const & m, level const & l, justification const & j) const { lean_assert(is_meta(m)); return assign(meta_id(m), l, j); }
+    substitution assign(level const & m, level const & l) { lean_assert(is_meta(m)); return assign(meta_id(m), l); }
 
     /** \brief Instantiate metavariables in \c e assigned in this substitution. */
     std::pair<expr, justification> instantiate_metavars(expr const & e) const;
@@ -56,8 +69,9 @@ public:
        \brief Similar to the previous function, but it compress the substitution.
        By compress, we mean, for any metavariable \c m reachable from \c e,
        if s[m] = t, and t has asssigned metavariables, then s[m] <- instantiate_metavars(t, s).
+       The updated substitution is returned.
     */
-    std::pair<expr, justification> d_instantiate_metavars(expr const & e);
+    std::tuple<expr, justification, substitution> updt_instantiate_metavars(expr const & e) const;
 
     /** \brief Instantiate level metavariables in \c l. */
     std::pair<level, justification> instantiate_metavars(level const & l) const;
@@ -68,7 +82,7 @@ public:
     */
     expr instantiate_metavars_wo_jst(expr const & e) const;
 
-    expr d_instantiate_metavars_wo_jst(expr const & e);
+    std::pair<expr, substitution> updt_instantiate_metavars_wo_jst(expr const & e) const;
 
     /** \brief Instantiate level metavariables in \c l, but does not return justification object. */
     level instantiate_metavars_wo_jst(level const & l) const;
diff --git a/src/tests/kernel/metavar.cpp b/src/tests/kernel/metavar.cpp
index 59f2bb7d7..f82ea7a48 100644
--- a/src/tests/kernel/metavar.cpp
+++ b/src/tests/kernel/metavar.cpp
@@ -75,7 +75,7 @@ static void tst1() {
     lean_assert(m1 != m2);
     expr f = Const("f");
     expr a = Const("a");
-    subst.assign(m1, f(a));
+    subst = subst.assign(m1, f(a));
     lean_assert(subst.is_assigned(m1));
     lean_assert(!subst.is_assigned(m2));
     lean_assert(*subst.get_expr(m1) == f(a));
@@ -91,8 +91,8 @@ static void tst2() {
     expr f = Const("f");
     expr g = Const("g");
     expr a = Const("a");
-    s.assign(m1, f(m2), mk_assumption_justification(1));
-    s.assign(m2, g(a),  mk_assumption_justification(2));
+    s = s.assign(m1, f(m2), mk_assumption_justification(1));
+    s = s.assign(m2, g(a),  mk_assumption_justification(2));
     lean_assert(check_assumptions(s.get_assignment(m1)->second, {1}));
     lean_assert(s.occurs(m1, f(m1)));
     lean_assert(s.occurs(m2, f(m1)));
@@ -104,13 +104,14 @@ static void tst2() {
     check_assumptions(p1.second, {1, 2});
     lean_assert(check_assumptions(s.get_assignment(m1)->second, {1}));
     lean_assert(p1.first == g(f(g(a))));
-    auto p2 = s.d_instantiate_metavars(g(m1, m3));
-    check_assumptions(p2.second, {1, 2});
-    std::cout << p2.first << "\n";
+    auto ts = s.updt_instantiate_metavars(g(m1, m3));
+    s = std::get<2>(ts);
+    check_assumptions(std::get<1>(ts), {1, 2});
+    std::cout << std::get<0>(ts) << "\n";
     std::cout << s << "\n";
     lean_assert(check_assumptions(s.get_assignment(m1)->second, {1, 2}));
-    lean_assert(p2.first == g(f(g(a)), m3));
-    s.assign(m3, f(m1, m2), mk_assumption_justification(3));
+    lean_assert(std::get<0>(ts) == g(f(g(a)), m3));
+    s = s.assign(m3, f(m1, m2), mk_assumption_justification(3));
     auto p3 = s.instantiate_metavars(g(m1, m3));
     lean_assert(check_assumptions(p3.second, {1, 2, 3}));
     std::cout << p3.first << "\n";
@@ -128,7 +129,7 @@ static void tst3() {
     expr y  = Const("y");
     expr a  = Const("a");
     expr b  = Const("b");
-    s.assign(m1, Fun({{x, Bool}, {y, Bool}}, f(y, x)));
+    s = s.assign(m1, Fun({{x, Bool}, {y, Bool}}, f(y, x)));
     lean_assert_eq(s.instantiate_metavars(m1(a, b, g(a))).first, f(b, a, g(a)));
     lean_assert_eq(s.instantiate_metavars(m1(a)).first, Fun({y, Bool}, f(y, a)));
     lean_assert_eq(s.instantiate_metavars(m1(a, b)).first, f(b, a));